joint carrier frequency offset and channel estimation for ...(ofdma) is a leading technology for...

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Joint Carrier Frequency Offset and Channel Estimation forUplink MIMO-OFDMA Systems Using Parallel Schmidt

Rao-Blackwellized Particle Filters

Kyeong Jin Kim, Man-On Pun, Ronald Iltis

TR2010-095 November 2010

AbstractJoint carrier frequency offset (CFO) and channel estimation for uplink MIMO-OFDMA sys-tems over time-varying channels is investigated. To cope with the prohibitive computa-tional complexity involved in estimating multiple CFOs and channels, pilot-assisted andsemi-blind schemes comprised of parallel Schmidt Extended Kalman filters (SEKFs) andSchmidt-Kalman Approximate Particle Filters (SK-APF) are proposed. In the SK-APF, aRao-Blackwellized particle filter (RBPF) is developed to first estimate the nonlinear statevariable, i.e. the desired user’s CFO, through the sampling-importance-resampling (SIRS)technique. The individual user channel responses are then updated via a bank of Kalmanfilters conditioned on the CFO sample trajectories. Simulation results indicate that theproposed schemes can achieve highly accurate CFO/channel estimates, and that the parti-cle filtering approach in the SK-APF outperforms the more conventional Schmidt ExtendedKalman Filter.

IEEE Transactions on Communications

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 9, SEPTEMBER 2010 2697

Joint Carrier Frequency Offset andChannel Estimation for Uplink

MIMO-OFDMA Systems Using ParallelSchmidt Rao-Blackwellized Particle Filters

Kyeong Jin Kim, Member, IEEE, Man-On Pun, Member, IEEE, and Ronald A. Iltis, Senior Member, IEEE

Abstract—Joint carrier frequency offset (CFO) and channelestimation for uplink MIMO-OFDMA systems over time-varyingchannels is investigated. To cope with the prohibitive compu-tational complexity involved in estimating multiple CFOs andchannels, pilot-assisted and semi-blind schemes comprised ofparallel Schmidt Extended Kalman filters (SEKFs) and Schmidt-Kalman Approximate Particle Filters (SK-APF) are proposed.In the SK-APF, a Rao-Blackwellized particle filter (RBPF) isdeveloped to first estimate the nonlinear state variable, i.e. the de-sired user’s CFO, through the sampling-importance-resampling(SIRS) technique. The individual user channel responses are thenupdated via a bank of Kalman filters conditioned on the CFOsample trajectories. Simulation results indicate that the proposedschemes can achieve highly accurate CFO/channel estimates, andthat the particle filtering approach in the SK-APF outperformsthe more conventional Schmidt Extended Kalman Filter.

Index Terms—Orthogonal frequency division multiple access(OFDMA), multiple-input multiple-output (MIMO), channel es-timation, frequency synchronization, Schmidt extended Kalmanfilter (SEKF), Rao-Blackwellized particle filter (RBPF).

I. INTRODUCTION

ORTHOGONAL Frequency Division Multiple Access(OFDMA) is a leading technology for broadband wire-

less networks[1]. In addition to its robustness to multipathfading and high spectral efficiency, OFDMA offers flexibilityin allocating subcarriers to different users based on qualityof service (QoS) requirements and channel conditions [2].Advances in multiple-input multiple-output (MIMO) tech-niques have led to considerable interest in MIMO-OFDMAin which substreams of a broadband source are transmittedover multiple antennas and subcarriers.

In this paper we consider an uplink MIMO-OFDMA systemover time-varying Rayleigh fading channels. In the uplink,

Paper approved by N. Al-Dhahir, the Editor for Space-Time, OFDMand Equalization of the IEEE Communications Society. Manuscript receivedJanuary 15, 2009; revised September 15, 2009 and January 15, 2010.

The work of M. O. Pun was done when he was a Croucher post-doctoralfellow at Princeton University, Princeton, NJ.

The work of R. A. Iltis was supported in part by a gift from Nokia.K. J. Kim was with Nokia Inc., 6000 Connection Drive, Irving, TX 75039

USA (e-mail: [email protected]).M. O. Pun is with Mitsubishi Electric Research Laboratories (MERL), 201

Broadway, Cambridge, MA 02139 USA (e-mail: [email protected]).R. A. Iltis is with the Department of Electrical and Computer Engineering,

University of California, Santa Barbara, CA 93106-9560 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TCOMM.2010.080310.090053

the MIMO-OFDMA system requires that active users must besynchronized in frequency in order to maintain orthogonality.1

Furthermore, accurate channel estimation is required whencoherent data detection is employed. Several schemes havebeen proposed to perform joint synchronization and channelestimation for uplink OFDMA systems, e.g. [4]–[8]. De-spite their good performance, these existing schemes assumethat channels experience block-fading and CFOs are time-invariant. However, such approaches may not be suitable foroutdoor channels with high user mobility where large Dopplershifts are present. In [9], an extended Kalman filter (EKF)-based scheme has been proposed to track the time variationsof CFOs and channels for single-user MIMO-OFDM systemsimpaired by multiple CFOs. By exploiting a set of pilots, [9]can provide accurate estimation of CFOs and channels. In[10], parallel EKFs are proposed to perform CFO estimationfor uplink OFDMA systems as well as multiple access inter-ference (MAI) cancellation. However, in addition to its highcomplexity required in computing the Kalman gain, the EKFapproach in [9] suffers from potential divergence caused bythe highly nonlinear dependence of the received signal on theCFOs [11]. To circumvent this problem, [11] has proposed aparticle filtering (PF)-based scheme [12] to track a single time-varying CFO for OFDM systems, assuming perfect channelinformation is available. Recently, joint frequency offset andchannel estimation has been proposed in [13] using Gauss-Hermite integration. Unfortunately, this approach is only validfor single-user systems.

In this work, both pilot-aided and semi-blind schemes areproposed to jointly estimate time-varying frequency offsetsand channels for multiuser uplink MIMO-OFDMA systems.The challenge in the uplink MIMO-OFDMA case can beappreciated by comparing with the simpler SISO-OFDMAcase. For instance, consider user separation techniques. ForSISO-OFDMA systems it has been shown that iterative directcompensation of one user’s CFO while treating other users’signals as interference can provide effective user separation[14]. However, for MIMO-OFDMA with each user distortedby multiple CFOs, direct compensation of only one CFO maynot lead to signal improvement due to the existence of otherCFOs of the same user. To circumvent this problem, two

1Interested readers should refer to [3] for a comprehensive tutorial onOFDMA synchronization.

0090-6778/10$25.00 c⃝ 2010 IEEE

2698 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 9, SEPTEMBER 2010

techniques are proposed here. First, to achieve user separationat low computational complexity, pilot-aided methods employthe Schmidt-Kalman filters to break down the multiuser CFOand channel estimation problem into separable sub-problemsby exploiting a few training blocks [15]. Each estimation sub-problem is then solved by a Rao-Blackwellized particle filterwhere the desired user’s CFO is estimated through sampling-importance-resampling (SIRS) and the channel response isupdated via a EKF conditioned on the CFO samples generatedby SIRS. By using the SIRS technique, the divergence problemdue to non-linearity introduced by the CFOs is reduced.Finally, since training blocks are usually only available atthe beginning of each data frame, a semi-blind approachexploiting the tentatively detected data symbols is developed toimprove estimation accuracy. Specifically a QR decomposition(QRD)-based detector is incorporated into the receivers toprovide reliable tentative data decisions. Simulation resultsconfirm the effectiveness of the proposed joint estimators forMIMO-OFDMA.

The rest of the paper is organized as follows. Section IIpresents the signal model for the MIMO-OFDMA system,followed by the suboptimal parallel Schmidt extended Kalmanfilter (SEKF) in Section III. Based on the parallel SEKF, aparallel Schmidt-Kalman approximate-Rao-Blackwellized par-ticle filter (SK-APF) is proposed in Section IV. The QRD-Mdetector semi-blind approach is given in Section V. Simulationresults are presented in Section VI and conclusions are givenin Section VII.

Notation: Vectors and matrices are denoted by boldfaceletters. ∥⋅∥ represents the Euclidean norm of the enclosedvector and ∣⋅∣ denotes the amplitude of the enclosed complex-valued quantity. Finally, we use E {⋅}, (⋅)∗, (⋅)𝑇 and (⋅)𝐻 forexpectation, complex conjugation, transposition and Hermitiantransposition.

II. SIGNAL AND CHANNEL MODELS FOR UPLINK

MIMO-OFDMA SYSTEMS

We consider an uplink OFDMA system with 𝑁 subcarriersand 𝐾 active users. The base station (BS) and each active userare equipped with 𝑀𝑟 and 𝑀𝑡 antennas, respectively. Eachuser is assigned 𝑁𝑘 exclusive subcarriers, where

∑𝐾𝑘=1𝑁𝑘 ≤

𝑁 . We denote the index set of carriers assigned to the 𝑘-

th user as ℐ𝑘△={𝑖1, 𝑖2, . . . , 𝑖𝑁𝑘} where 1 ≤ 𝑖𝑙 ≤ 𝑁 for

𝑙 = 1, 2, ⋅ ⋅ ⋅ , 𝑁𝑘. The proposed algorithms are applicable toany carrier assignment scheme (CAS). Note that even thoughwe are using orthogonal subcarriers, self-interference and MAIare inevitable due to CFOs [16]. Denote by 𝒅𝑝𝑘(𝑛) ∈ ℂ𝑁

the data symbols transmitted by the 𝑘-th user from the 𝑝-th transmit antenna over the 𝑛-th OFDMA block. For con-venience, we assume that the data symbols are taken fromthe same complex-valued finite alphabet and independentlyidentically distributed (i.i.d). The 𝑖-th entry of 𝒅𝑝𝑘(𝑛), 𝑑

𝑝𝑘,𝑖(𝑛),

is non-zero if and only if 𝑖 ∈ ℐ𝑘. Next, 𝒅𝑝𝑘(𝑛) is converted tothe corresponding time-domain vector by an 𝑁 -point inversediscrete Fourier transform (IDFT):

𝒅𝑝𝑘(𝑛) = 𝑾𝐻𝒅𝑝𝑘(𝑛), (1)

where 𝑾𝐻 is the IDFT matrix. To prevent inter-symbolinterference (ISI), a cyclic prefix (CP) of 𝑁𝑔 symbols is

appended in front of each IDFT output block. The resultingvector of length 𝑁𝑔𝑑 = 𝑁 +𝑁𝑔 is digital-to-analog convertedby a pulse-shaping filter with a finite support on [0, 𝑇𝑑) where𝑇𝑑 = 𝑁𝑔𝑑𝑇𝑠 with 1/(𝑁𝑇𝑠) being the subcarrier spacing.Finally, the analog signal from the pulse-shaping filter istransmitted from the 𝑝-th antenna over the channel.

The channel between the 𝑝-th transmit antenna of the 𝑘-th user and the 𝑞-th receive antenna of the BS during the𝑛-th block, {ℎ𝑝,𝑞𝑘,𝑙 (𝑛), 0 ≤ 𝑙 ≤ 𝐿𝑝,𝑞𝑘 − 1}, is modeled as atapped delay line with 𝐿𝑝,𝑞𝑘 being the channel order. Since𝐿𝑝,𝑞𝑘 is generally unknown, in practice we replace 𝐿𝑝,𝑞𝑘 with𝐿𝑓 for all users and antenna pairs. For the widely acceptedITU channel model, an upper bound for 𝐿𝑓 can be specified.We assume that the length of the CP is sufficient to comprisethe maximum path delay, i.e., 𝐿𝑓 ≤ 𝑁𝑔 . Furthermore, weassume that {ℎ𝑝,𝑞𝑘,𝑙 (𝑛)} is constant over a OFDMA block 𝑛 oflength 𝑁𝑔𝑑𝑇𝑠 s but varies between blocks.

A. Signal Model in the Presence of CFOs

Due to the Doppler effect and oscillator mismatch betweentransmitter and receiver pairs, the received signal is usuallydistorted by carrier frequency offsets [17]. Since the antennaseparation on each user’s mobile terminal is generally muchsmaller than that on the BS, it can be reasonably assumedthat the CFO between transmit antennas of the same userand a specific receive antenna on the BS is identical. That is,

{𝛿𝑓 𝑞𝑘 = 𝛿𝑓𝑝,𝑞𝑘 , ∀𝑝}. Let 𝜀𝑞𝑘△=𝛿𝑓 𝑞𝑘𝑁𝑇𝑠 be the normalized carrier

frequency offset with respect to (w.r.t.) the carrier spacing1/(𝑁𝑇𝑠) between transmit antennas of the 𝑘-th user and the𝑞-th receive antenna of the BS. In the presence of CFOs,the received vector signal after removing the guard intervalbecomes [6], [14], [17], [18]

𝒓𝑞(𝑛) =

𝐾∑𝑘=1

Δ(𝜀𝑞𝑘(𝑛))

𝑀𝑡∑𝑝=1

𝑫𝑝𝑘(𝑛)𝒉𝑝,𝑞𝑘 (𝑛) + 𝒗𝑞(𝑛),

=

𝐾∑𝑘=1

�̃�𝑞𝜀,𝑘(𝑛)𝒉𝑞𝑘(𝑛) + 𝒗𝑞(𝑛), (2)

where

𝒉𝑝,𝑞𝑘 (𝑛)△=[ℎ𝑝,𝑞𝑘,0(𝑛), ℎ

𝑝,𝑞𝑘,1(𝑛), . . . , ℎ

𝑝,𝑞𝑘,𝐿𝑓−1(𝑛)

]𝑇,

𝑫𝑝𝑘(𝑛)△=

⎡⎢⎢⎢⎣

𝑑𝑝𝑘,0(𝑛) 𝑑𝑝𝑘,𝑁−1(𝑛) . . . 𝑑𝑝𝑘,𝑁−𝐿𝑓+1(𝑛)

𝑑𝑝𝑘,1(𝑛) 𝑑𝑝𝑘,0(𝑛) . . . 𝑑𝑝𝑘,𝑁−𝐿𝑓+2(𝑛)...

... . . . . . .𝑑𝑝𝑘,𝑁−1(𝑛) 𝑑𝑝𝑘,𝑁−2(𝑛) . . . 𝑑𝑝𝑘,𝑁−𝐿𝑓

(𝑛)

⎤⎥⎥⎥⎦ ,

Δ(𝜀𝑞𝑘(𝑛))△=𝑒𝑗𝜃

𝑞𝑘(𝑛)diag(1, 𝑒𝑗

2𝜋𝜀𝑞𝑘(𝑛)

𝑁 , .., 𝑒𝑗2𝜋(𝑁−1)𝜀

𝑞𝑘(𝑛)

𝑁 ),

𝜃𝑞𝑘(𝑛)△=2𝜋

𝑛−1∑𝑙=0

𝜀𝑞𝑘(𝑙),

𝒉𝑞𝑘(𝑛)△=[𝒉1,𝑞𝑘 (𝑛)𝑇 ,𝒉2,𝑞

𝑘 (𝑛)𝑇 , ..,𝒉𝑀𝑡,𝑞𝑘 (𝑛)𝑇

]𝑇,

�̃�𝑞𝜀,𝑘(𝑛)△=[Δ(𝜀𝑞𝑘(𝑛))𝑫

1𝑘(𝑛), ..,Δ(𝜀𝑞𝑘(𝑛))𝑫

𝑀𝑡

𝑘 (𝑛)].

(3)

Recall that the CP length equals the channel length plus timingoffset. Under such an assumption, the timing errors do not

KIM et al.: JOINT CARRIER FREQUENCY OFFSET AND CHANNEL ESTIMATION FOR UPLINK MIMO-OFDMA SYSTEMS 2699

explicitly appear in the received signal model [14]. Thus, wehave suppressed the timing errors in (2).

Several approaches have been proposed to model thetime-varying channels and frequency offsets in mobile en-vironments. Since we assume a normalized Doppler spread𝑓𝐷𝑇𝑑 ≪ 1,we adopt the following first-order autoregressive(AR) parametric model widely used in [10], [13], [18]–[20]to characterize the time-varying frequency offset and channelresponses.

𝜀𝑞𝑘(𝑛) = 𝛼𝑞𝑘,𝜀𝜀𝑞𝑘(𝑛− 1) + 𝑤𝑞𝑘,𝜀(𝑛),

𝒉𝑞𝑘(𝑛) = 𝛼𝑞𝑘,ℎ𝒉𝑞𝑘(𝑛− 1) +𝒘𝑞𝑘,ℎ(𝑛), (4)

where 𝑤𝑞𝑘,𝜀(𝑛) ∼ 𝒩 (𝑤𝑞𝑘,𝜀(𝑛); 0, 𝜂𝑞𝑘,𝜀) and 𝒘𝑞𝑘,ℎ(𝑛) ∼

𝒞𝒩 (𝒘𝑞𝑘,ℎ(𝑛);0, 𝜂𝑞𝑘,ℎ𝑰𝑀𝑡𝐿𝑓

). Furthermore, we assume that∣∣∣𝛼𝑞𝑘,𝜀∣∣∣ < 1 and

∣∣∣𝛼𝑞𝑘,ℎ∣∣∣ < 1. Note that we assume independent

fading across transmit antennas and multipaths. The time-variation in CFO in (4) arises from (a) local oscillator insta-bility due to temperature/voltage variations and (b) changesin relative platform Doppler velocity. In the next section, wepropose a pilot-aided parallel Schmidt extended Kalman filterapproach to estimate 𝜀𝑞𝑘(𝑛) and 𝒉𝑞𝑘(𝑛) based on the received

signal 𝒓𝑞(𝑛), assuming exact knowledge of {𝑫𝑝𝑘(𝑛)},{𝛼𝑞𝑘,𝜀

},

and{𝛼𝑞𝑘,ℎ

}in the BS.

III. PARALLEL SCHMIDT KALMAN FILTER FOR JOINT

CHANNEL AND CFO ESTIMATION

A. Kalman Filter FormulationTo facilitate the derivation of the Schmidt Kalman filter,

we need to convert the complex signal model developed inthe previous section into a real-valued form. We begin bydefining the following real-valued quantities.

ℍ(𝜀(𝑛))△=ℍ

𝑞𝑘(𝜀

𝑞𝑘(𝑛))

=

[Re{�̃�𝑞

𝜀,𝑘(𝑛)} −Im{�̃�𝑞𝜀,𝑘(𝑛)}

Im{�̃�𝑞𝜀,𝑘(𝑛)} Re{�̃�𝑞

𝜀,𝑘(𝑛)}

],

𝒇(𝑛)△=𝒇𝑞

𝑘 (𝑛) = [Re{𝒉𝑞𝑘(𝑛)}𝑇 , Im{𝒉𝑞

𝑘(𝑛)}𝑇 ]𝑇 ,ℍ∖𝑘(𝜀∖𝑘(𝑛)) =[ℍ

𝑞1(𝜀1(𝑛)), ..,ℍ

𝑞𝑘−1(𝜀𝑘−1(𝑛)),ℍ

𝑞𝑘+1(𝜀𝑘+1(𝑛)), ..ℍ

𝑞𝐾(𝜀𝐾(𝑛))

],

𝒇∖𝑘(𝑛) = [𝒇𝑞1 (𝑛)

𝑇 , .., 𝒇𝑞𝑘−1(𝑛)

𝑇 ,𝒇𝑞𝑘+1(𝑛)

𝑇 , .., 𝒇𝑞𝐾(𝑛)𝑇 ]𝑇 ,

𝒛(𝑛)△=𝒛𝑞

𝑘(𝑛) ∼ 𝒩 (𝒛𝑞𝑘(𝑛);0, 𝑁0/𝑇𝑠𝑰2𝑁 ),

(5)

where the subscript ∖𝑘 stands for the exclusion of the parame-ters associated with the 𝑘-th user. For example, 𝜀∖𝑘(𝑛) denotesall 𝜀𝑖(𝑛)s except 𝜀𝑘(𝑛).

Utilizing the real-valued quantities defined above, (2) can berewritten as the following real-valued system and observationequations:

𝒚(𝑛) = ℍ(𝜀(𝑛))𝒇(𝑛) +ℍ∖𝑘(𝜀∖𝑘(𝑛))𝒇∖𝑘(𝑛) + 𝒛(𝑛),

𝒇(𝑛) = 𝑨𝑞𝑘,ℎ𝒇(𝑛− 1) +𝒘𝑟𝑘,ℎ(𝑛),

𝜀(𝑛) = 𝛼𝑞𝑘,𝜀𝜀(𝑛− 1) + 𝑤𝑞𝑘,𝜀(𝑛), (6)

where 𝑨𝑞𝑘,ℎ = [𝑰2 ⊗ 𝛼𝑞𝑘,ℎ𝑰𝑀𝑡𝐿𝑓], and 𝒘𝑟𝑘,ℎ(𝑛) is similarly

the real/imaginary partitioned version of the process noisein (4). Note that 𝒚(𝑛) in (6) corresponds to the receivedvector only from the 𝑞-th antenna and 𝒇(𝑛) and ℍ(𝜀(𝑛))

are quantities related to the 𝑘-th user as received at antenna𝑞. It should be emphasized that although the 𝑘-th user usesorthogonal sets of subcarriers in ℐ𝑘, self-interference and MAIoccur due to CFOs [16]. Thus, vectors ℍ(𝜀(𝑛))𝒇(𝑛) andℍ∖𝑘(𝜀∖𝑘(𝑛))𝒇∖𝑘(𝑛) cannot remain orthogonal. The EKF canbe employed to jointly estimate 𝒇𝑞𝑘 (𝑛) and 𝜖𝑞𝑘(𝑛) by concate-nating all those vectors for 𝑘 = 1, 2, . . .𝐾 . However, suchEKF approaches are susceptible to divergence problems dueto the nonlinear nature of CFO [9], [10]. Furthermore, directcomputation of the Kalman gain for (6) using the concatenatedstate vector of all users is highly inefficient. In the following,we propose to use the suboptimal parallel Schmidt extendedKalman filters to develop a feasible estimator.

B. Parallel Schmidt Extended Kalman Filter (SEKF)

Following the notation in [15], we propose a parallel bankof 𝐾 Schmidt extended Kalman filters (SEKFs) with eachSEKF related to one desired user. Without loss of generality,we assume that the 𝑘-th user is the desired user for the 𝑘-thSEKF. In the 𝑘-th SEKF, we divide the state vector into theessential state vector (ESV) related to the 𝑞-th antenna of the𝑘-th user,

𝒙(𝑛)△= [𝜀(𝑛),𝒇(𝑛)𝑇 ]𝑇 , (7)

and the nuisance state vector (NSV) 𝒙∖𝑘(𝑛) for the interferingusers. Note that 𝒙(𝑛) in (7) is understood to represent 𝒙𝑞𝑘(𝑛).

Substituting (7) and 𝒙∖𝑘(𝑛) into (6) and applying a first-order approximation, we can obtain the following new pair oflinearized system and observation equations as

𝒚(𝑛) ≈ ℍ(𝜀(𝑛∣𝑛− 1))𝒇(𝑛∣𝑛− 1)

+ℍ∖𝑘(𝜀∖𝑘(𝑛∣𝑛− 1))𝒇∖𝑘(𝑛∣𝑛− 1)

+[𝑱(𝑛) 𝑱∖𝑘(𝑛)

] [ 𝒙(𝑛)− �̂�(𝑛∣𝑛− 1)𝒙∖𝑘(𝑛)− �̂�∖𝑘(𝑛∣𝑛− 1)

]+ 𝒛(𝑛),

𝒙(𝑛) = 𝑨𝒙(𝑛− 1) +𝒘(𝑛),

𝒙∖𝑘(𝑛) = 𝑨∖𝑘𝒙∖𝑘(𝑛− 1) +𝒘∖𝑘(𝑛),

(8)

where 𝑨 = blkdiag(𝛼𝑞𝑘,𝜀,𝑨

𝑞𝑘,ℎ

)and 𝒙∖𝑘(𝑛) denotes all

state vectors excluding 𝒙𝑘(𝑛) of the desired user. The noiseterm in (8) is given as 𝒘(𝑛) ∼ 𝒩 (𝒘(𝑛);0,𝑸) with 𝑸 =

blkdiag(𝜂𝑞𝑘,𝜀, 𝑰2 ⊗ 𝜂𝑞𝑘,ℎ𝑰𝑀𝑡𝐿𝑓

/2). Similarly we can compute

𝑨∖𝑘, 𝒘∖𝑘(𝑛) ∼ 𝒩 (𝒘∖𝑘(𝑛);0,𝑸∖𝑘) and 𝑸∖𝑘. Furthermore,the Jacobian matrix 𝑱(𝑛) in (8) is the gradient w.r.t. the statevector of the nonlinear measurement function and equals

𝑱(𝑛)△= [𝑱𝜀(𝑛) 𝑱ℎ(𝑛)] ∈ ℝ

2𝑁×(2𝑀𝑡𝐿𝑓+1), (9)

See (10) at the top of the next page.The Jacobian 𝑱∖𝑘(𝑛) can be similarly obtained and reads

𝑱∖𝑘(𝑛) = [𝑱1(𝑛), ..,𝑱𝑘−1(𝑛),𝑱𝑘+1(𝑛), ..,𝑱𝐾(𝑛)]

∈ ℝ2𝑁×(𝐾−1)(2𝑀𝑡𝐿𝑓+1).

(11)

For the conventional EKF, the covariance matrix 𝑷 (𝑛∣𝑛−1)


𝑱𝜀(𝑛) =

[ −Im{ΛΔ(𝜀(𝑛∣𝑛− 1))𝑫𝑞𝜀,𝑘(𝑛)},−Re{ΛΔ(𝜀(𝑛∣𝑛− 1))𝑫𝑞𝜀,𝑘(𝑛)}Re{ΛΔ(𝜀(𝑛∣𝑛− 1))𝑫𝑞𝜀,𝑘(𝑛)},−Im{ΛΔ(𝜀(𝑛∣𝑛− 1))𝑫𝑞𝜀,𝑘(𝑛)}

]× 𝒇(𝑛∣𝑛− 1),

𝑱ℎ(𝑛) = ℍ(𝜀(𝑛∣𝑛− 1)),Λ△=2𝜋

𝑁diag (0, 1/𝑁, ..(𝑁 − 1)/𝑁) .

(10)

and the Kalman gain matrix 𝑲(𝑛) take the following forms.

𝑷 (𝑛∣𝑛− 1) =[𝑷𝑘,𝑘(𝑛∣𝑛− 1) 𝑷𝑘,∖𝑘(𝑛∣𝑛− 1)𝑷∖𝑘,𝑘(𝑛∣𝑛− 1) 𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)

],

𝑲(𝑛) =

[𝑲𝑘(𝑛)𝑲∖𝑘(𝑛)

]=

[𝑷𝑘,𝑘(𝑛∣𝑛− 1)𝑱(𝑛)𝑇 + 𝑷𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇 ,𝑷∖𝑘,𝑘(𝑛∣𝑛− 1)𝑱(𝑛)𝑇 + 𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇

]𝒜,(12)

where

(𝒜)−1 =

𝑱(𝑛)𝑷𝑘,𝑘(𝑛∣𝑛− 1)𝑱(𝑛)𝑇 + 𝑱(𝑛)𝑷𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇+

𝑱∖𝑘(𝑛)𝑷∖𝑘,𝑘(𝑛∣𝑛− 1)𝑱(𝑛)𝑇+

𝑱∖𝑘(𝑛)𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇 +𝑁0/𝑇𝑠𝑰.(13)

In (12), 𝑲(𝑛) is the total EKF Kalman gain for joint trackingof all uplink users.

In lieu of computing the composite 𝑲(𝑛) in (12), the SEKFsets the NSV part of the Kalman gain 𝑲∖𝑘(𝑛) to zero, whichresults in

�̂�(𝑛) = [𝑲𝑘,SKF(𝑛)𝑇 ,0𝑇 ]𝑇 . (14)

Approximation (14) is exact when estimation errors betweenthe 𝑘-th user and its interfering users are uncorrelated andthe error covariance of the interfering users is zero, i.e.𝑷∖𝑘,𝑘(𝑛∣𝑛− 1) = 0 and 𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1) = 0.

Now the estimate for the ESV becomes

𝑲𝑘,SKF(𝑛) =(𝑷𝑘,𝑘(𝑛∣𝑛− 1)𝑱(𝑛)𝑇 +𝑷𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)

𝑇)𝒜,

�̂�(𝑛∣𝑛) = �̂�(𝑛∣𝑛− 1) +𝑲𝑘,SKF(𝑛) (𝒚(𝑛)−ℍ(𝜀(𝑛∣𝑛− 1))𝒇 (𝑛∣𝑛− 1) −ℍ∖𝑘(𝜀∖𝑘(𝑛∣𝑛− 1))𝒇∖𝑘(𝑛∣𝑛− 1)

),

(15)

where the predicted estimate �̂�(𝑛+1∣𝑛) and the correspondingerror covariance matrix 𝑷𝑘,𝑘(𝑛+1∣𝑛) for the desired user arecomputed as follows.

�̂�(𝑛+ 1∣𝑛) = 𝑨�̂�(𝑛∣𝑛),𝑷𝑘,𝑘(𝑛+ 1∣𝑛) = 𝑨𝑷𝑘,𝑘(𝑛∣𝑛)(𝑨)𝑇 +𝑸. (16)

To evaluate (16), we need to compute the joint NSV andESV error covariance matrix based on (15), 𝑷 (𝑛∣𝑛) whichfollowing [15] is

𝑷 (𝑛∣𝑛) =

[𝑷𝑘,𝑘(𝑛∣𝑛) 𝑷𝑘,∖𝑘(𝑛∣𝑛)𝑷∖𝑘,𝑘(𝑛∣𝑛) 𝑷∖𝑘,∖𝑘(𝑛∣𝑛)

], (17)

with each sub-matrix given by

𝑷𝑘,𝑘(𝑛∣𝑛) =ℬ𝑷𝑘,𝑘(𝑛∣𝑛− 1)(ℬ)𝑇 − ℬ𝑷𝑘,∖𝑘(𝑛∣𝑛)(𝒞)𝑇 − 𝒞𝑷∖𝑘,𝑘(𝑛∣𝑛)(ℬ)𝑇+𝒞𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)(𝒞)𝑇 +

𝑁0

𝑇𝑠𝑲𝑘,SKF(𝑛)𝑲𝑘,SKF(𝑛)

𝑇 ,

𝑷𝑘,∖𝑘(𝑛∣𝑛) = 𝑷∖𝑘,𝑘(𝑛∣𝑛)𝑇= ℬ𝑷𝑘,∖𝑘(𝑛∣𝑛− 1)− 𝒞𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1),

𝑷∖𝑘,∖𝑘(𝑛∣𝑛) = 𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1),(18)

where ℬ△=(𝑰 − 𝑲𝑘,SKF(𝑛)𝑱(𝑛)) and 𝒞△

=𝑲𝑘,SKF(𝑛)𝑱∖𝑘(𝑛).Note that the error covariance for the interfering users𝑷∖𝑘,∖𝑘(𝑛∣𝑛) is not updated by the 𝑘-th SEKF.

To summarize the SEKF, the 𝑘-th filter produces {�̂�𝑘(𝑛+1∣𝑛),𝑷𝑘,𝑘(𝑛+1∣𝑛)} by exploiting the predictions {�̂�𝑘(𝑛∣𝑛−1), �̂�∖𝑘(𝑛∣𝑛−1),𝑷𝑘,𝑘(𝑛∣𝑛−1),𝑷∖𝑘,∖𝑘(𝑛∣𝑛−1)}. If the crosserror covariance is zero, i.e. 𝑷𝑘,∖𝑘(𝑛∣𝑛 − 1) = 0, then wehave the so-called inflated covariance system [21]. It is worthnoting that 𝑷𝑘,∖𝑘(𝑛∣𝑛) = 0 only when 𝑷𝑘,∖𝑘(𝑛∣𝑛 − 1) = 0and 𝑷∖𝑘,∖𝑘(𝑛∣𝑛−1) = 0. In general, we have 𝑷𝑘,∖𝑘(𝑛∣𝑛) ∕= 0due to the error covariance of the interfering users.

IV. PARALLEL APPROXIMATE-RAO-BLACKWELLIZED

PARTICLE FILTER

A Rao-Blackwellized particle filter using the first-orderapproximation to the observation is proposed for each SEKFwith structure suggested in [12], [22]–[24]. The state vectoris factorized into two terms related to the channel coefficients𝒇(𝑛) and the CFO state variable 𝜀(𝑛), respectively.

Let 𝒇(𝑛) be the channel vector and 𝜖𝑛𝑖 = {𝜖𝑖(𝑛), 𝜖𝑖(𝑛 −1) . . . 𝜖𝑖(0)} be the 𝑖-th trajectory of frequency offset particlesfor the 𝑘-th user and the 𝑞-th receive antenna with 𝑘 and𝑞 again suppressed for clarity. We define the cumulative

observation sequences as 𝒚𝑛△={𝒚(𝑛),𝒚(𝑛 − 1), ⋅ ⋅ ⋅ ,𝒚(0)}.

The importance sampling estimate of 𝑝(𝒇(𝑛)∣𝒚𝑛) is given by[24]

𝑝(𝒇(𝑛)∣𝒚𝑛) =𝑁𝑝∑𝑖=1

𝛽𝑖(𝑛)𝑝(𝒇(𝑛)∣𝒚𝑛, 𝜖𝑛𝑖 ). (19)

We assume 𝑁𝑝 particles in the Rao-Blackwellization. In [24],the importance sampling weight for unbiased estimation isgiven by

𝛽𝑖(𝑛) =𝑝(𝜖𝑛𝑖 ∣𝒚𝑛)

𝜋(𝜖𝑖(𝑛)∣𝒚𝑛, 𝜖𝑛−1𝑖 )𝜋(𝜖𝑛−1

𝑖 ∣𝒚𝑛−1), (20)

where 𝜋 (⋅) denotes the sampling density. Note that theprior sampling density 𝜋(𝜖𝑛−1

𝑖 ∣𝒚𝑛−1) is a causal functionof measurements up to time 𝑛 − 1. The minimum variancechoice [25] for the sample density of current sample 𝜖𝑖(𝑛) is𝜋(𝜖𝑖(𝑛)∣𝒚𝑛, 𝜖𝑛−1

𝑖 ) = 𝑝(𝜖𝑖(𝑛)∣𝒚𝑛, 𝜖𝑛−1𝑖 ), i.e. the true density

function of the CFO given the past trajectory 𝜖𝑛−1𝑖 and


cumulative measurements 𝒚𝑛. The conditional distributionunder this Rao-Blackwellization of 𝒇(𝑛) is thus given by :

𝑝(𝒇(𝑛)∣𝒚𝑛, 𝜖𝑛𝑖 ) = 𝒩 (𝒇(𝑛);𝒇𝑖(𝑛∣𝑛),𝑷𝑘,𝑘,𝑖(𝑛∣𝑛)), (21)

where 𝒇𝑖(𝑛∣𝑛),𝑷𝑘,𝑘,𝑖(𝑛∣𝑛) are the Kalman filter estimate andcovariance using the particular CFO trajectory 𝜖𝑛𝑖 , respectively.These quantities will be approximated using the Schmidt-Kalman filter for the multi-user case.

To utilize Rao-Blackwellization, we rewrite the linearizedsystem in (8) by removing the terms related to 𝑱(𝑛), i.e. onlythe first-order linearization w.r.t. NSV is employed.

𝒚(𝑛) ≈ ℍ(𝜀(𝑛))𝒇(𝑛)+

ℍ∖𝑘(𝜀∖𝑘(𝑛∣𝑛− 1))𝒇∖𝑘(𝑛∣𝑛− 1)+

𝑱∖𝑘(𝑛)[𝒙∖𝑘(𝑛)− �̂�∖𝑘(𝑛∣𝑛− 1)

]+ 𝒛′(𝑛).

(22)

Using the Kalman filter quantities yields

𝜋(𝜖𝑖(𝑛)∣𝒚𝑛, 𝜖𝑛−1𝑖 ) ∝ 𝑝(𝒚(𝑛)∣𝒚𝑛−1, 𝜖𝑛𝑖 )𝑝(𝜖𝑖(𝑛)∣𝜖𝑖(𝑛− 1)) =

𝒩 (𝒚(𝑛);𝒚𝑖(𝑛∣𝑛− 1),Σ𝑖(𝑛∣𝑛− 1))𝒩 (𝜖𝑖(𝑛);𝛼𝑞𝑘,𝜖𝜖𝑖(𝑛− 1), 𝜂𝑞

𝑘,𝜖),(23)

where

𝒚𝑖(𝑛∣𝑛− 1) =

ℍ(𝜖𝑖(𝑛))𝒇𝑖(𝑛∣𝑛− 1) +ℍ∖𝑘(𝜀∖𝑘(𝑛∣𝑛− 1))𝒇∖𝑘(𝑛∣𝑛− 1),

Σ𝑖(𝑛∣𝑛− 1) =

𝑁0/𝑇𝑠𝑰 +ℍ(𝜖𝑖(𝑛))𝑷𝑘,𝑘,𝑖(𝑛∣𝑛− 1)ℍ(𝜖𝑖(𝑛))𝑇+

𝑱∖𝑘(𝑛)𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇 .(24)

The importance weight corresponding to the minimum-variance sampling distribution is [25]

𝛽𝑖(𝑛) =1

𝑐1𝑝(𝒚(𝑛)∣𝜖𝑛−1

𝑖 ,𝒚𝑛−1)𝛽𝑖(𝑛− 1), (25)

where 𝑐1 is a constant. The density function in (25) can beapproximated using the samples 𝜖𝑗(𝑛) as

𝑝(𝒚(𝑛)∣𝜖𝑛−1𝑖 ,𝒚𝑛−1) =∫

𝑝(𝒚(𝑛)∣𝜀(𝑛), 𝜖𝑛−1𝑖 ,𝒚𝑛−1)𝑝(𝜀(𝑛)∣𝜖𝑛−1

𝑖 ,𝒚𝑛−1)𝑑𝜀(𝑛) ≈

1

𝑐2

𝑁𝑝∑𝑗=1

𝒩(𝒚(𝑛);ℍ(𝜖𝑗(𝑛))𝒇𝑖(𝑛∣𝑛− 1)+

ℍ∖𝑘(𝜀∖𝑘(𝑛∣𝑛− 1) 𝒇∖𝑘(𝑛∣𝑛− 1),Σ𝑖(𝑛∣𝑛− 1))×

𝒩(𝜖𝑗(𝑛);𝛼

𝑞𝑘,𝜖𝜖𝑖(𝑛− 1), 𝜂𝑞𝑘,𝜖

),

(26)

where the constant 1/𝑐2 normalizes (26) to a valid probabilitydensity function. The weight 𝛽𝑖(𝑛) is independent of thecurrent sample 𝜖𝑖(𝑛), but dependent on the past trajectory𝜖𝑛−1𝑖 .

A. Sampling With M-Statistics

To generate the frequency offset sample {𝜖𝑖(𝑛)}, we assumein the following that we have 𝜖𝑛−1

𝑖 , {𝒇𝑖(𝑛 − 1∣𝑛 − 1)}, and{𝑷𝑘,𝑘,𝑖(𝑛−1∣𝑛−1)}. Using the dynamic model, we generate

a tentative sample that will be updated later after the samplingprocess :

𝜖𝑖(𝑛) = 𝛼𝑞𝑘,𝜖𝜖𝑖(𝑛− 1) + 𝑤𝑞𝑘,𝜖. (27)

Conditioned on 𝜖𝑖(𝑛), we compute 𝒇𝑖(𝑛∣𝑛 − 1) and𝑷𝑘,𝑘,𝑖(𝑛∣𝑛− 1) via (16) with 𝒇𝑖(𝑛− 1∣𝑛− 1) and 𝑷𝑘,𝑘,𝑖(𝑛−1∣𝑛− 1) computed from the trajectory 𝜖𝑛−1

𝑖 .The CFO uncertainty region is approximated by a 2𝑀 + 1

point grid on the interval [−𝑀Δ𝜖,𝑀Δ𝜖) with 𝛿𝑚 = 𝑚Δ𝜖,where𝑚 = −𝑀,−𝑀+1, ⋅ ⋅ ⋅ ,𝑀 . Next, we compute the sam-pling distributions at the grid points using a new measurement𝒚(𝑛) as follows :

𝜋𝑖(𝑚)△=𝒩 (𝒚(𝑛);𝒚(𝑖,𝑚)(𝑛∣𝑛− 1),Σ(𝑖,𝑚)(𝑛∣𝑛− 1))×

𝒩 (𝛿𝑚;𝛼𝑞𝑘,𝜖𝜖𝑖(𝑛− 1), 𝜂𝑞𝑘,𝜖),

(28)

where

𝒚(𝑖,𝑚)(𝑛∣𝑛− 1) = ℍ(𝛿𝑚)𝒇𝑖(𝑛∣𝑛− 1)

+ ℍ∖𝑘(𝜀∖𝑘(𝑛∣𝑛− 1))𝒇∖𝑘(𝑛∣𝑛− 1)

and

Σ(𝑖,𝑚)(𝑛∣𝑛− 1) = 𝑁0/𝑇𝑠𝑰

+ ℍ(𝛿𝑚)𝑷𝑘,𝑘,(𝑖,𝑚)(𝑛∣𝑛− 1)ℍ(𝛿𝑚)𝑇

+ 𝑱∖𝑘(𝑛)𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇 .

In this equation, 𝑷𝑘,𝑘,(𝑖,𝑚)(𝑛∣𝑛− 1) is the corresponding co-variance for particle 𝑖 and grid point 𝛿𝑚. Upon computing all𝜋𝑖(𝑚), we choose a random value �̃� satisfying the followinginequality:

�̃�∑𝑙=−𝑀

𝜋𝑖(𝑙)∑𝑀𝑙′=−𝑀 𝜋𝑖(𝑙′)

> 𝑢 >

�̃�−1∑𝑙=−𝑀


, (29)

where 𝑢 is a uniform r.v. on [0, 1]. With �̃� obtained above,we select 𝜖𝑖(𝑛) = 𝛿�̃� and update 𝜖𝑛𝑖 = {𝜖𝑖(𝑛), 𝜖𝑛−1

𝑖 }.Finally, the resulting conditional distribution given in (21)

can be computed as

𝑝(𝒇(𝑛)∣𝒚𝑛, 𝜖𝑛𝑖 ) = 𝒩 (𝒇(𝑛);𝒇𝑖(𝑛∣𝑛),𝑷𝑘,𝑘,𝑖(𝑛∣𝑛)), (30)

where

𝒇𝑖(𝑛∣𝑛) = 𝒇𝑖(𝑛∣𝑛− 1) +𝑲𝑖(𝑛)(𝒚(𝑛) − 𝒚𝑖(𝑛∣𝑛− 1)),

𝑲𝑖(𝑛) =(𝑷𝑘,𝑘,𝑖(𝑛∣𝑛− 1)𝑱𝑖(𝑛)

𝑇 + 𝑷(𝑘,𝑖),∖𝑘(𝑛∣𝑛− 1)𝑱∖𝑘(𝑛)𝑇)𝒜𝑖(31)

The matrices 𝑷(𝑘,𝑖),∖𝑘(𝑛∣𝑛 − 1), 𝑱𝑖(𝑛), and 𝒜𝑖 correspondto 𝑷𝑘,∖𝑘(𝑛∣𝑛 − 1), 𝑱(𝑛), and 𝒜 for the 𝑖-th particle. Theseterms are defined in equations (18),(9), and (13). However, theJacobian 𝑱𝑖(𝑛) reduces to ℍ(𝜖𝑖(𝑛)) for the particle filter case.These particle dependent terms are updated independently bythe 𝑘 users. Thus, the unconditional distribution is given by

𝑝(𝒇(𝑛)∣𝒚𝑛) =𝑁𝑝∑𝑖=1

𝛽𝑖(𝑛)𝒩 (𝒇(𝑛);𝒇𝑖(𝑛∣𝑛),𝑷𝑘,𝑘,𝑖(𝑛∣𝑛)).(32)

Since the proposed scheme consists of suboptimal parallelfilters, it is referred to as the Schmidt-Kalman approximate-Rao-Blackwellized particle filter (SK-APF) in the sequel. SK-APF is summarized in Algorithm 1.


Algorithm 1 Parallel Schmidt-Kalman Approximate-Rao-Blackwellized Particle Filter

1. Given 𝜖𝑛−1𝑖 , 𝒇𝑖(𝑛− 1∣𝑛− 1),𝑷𝑘,𝑘,𝑖(𝑛− 1∣𝑛− 1) for the 𝑘-th user

(a) Generate a sample 𝜖𝑖(𝑛) = 𝛼𝑞𝑘,𝜀𝜖𝑖(𝑛− 1) + 𝑤𝑞

𝑘,𝜀.

(b) Compute interfering channel prediction 𝒇∖𝑘(𝑛∣𝑛− 1) and covariance𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1), ∀𝑘.

𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1) = 𝑨∖𝑘𝑷∖𝑘,∖𝑘(𝑛− 1∣𝑛− 1)𝑨𝑇∖𝑘 +𝑸∖𝑘

𝑷∖𝑘,∖𝑘(𝑛∣𝑛) = 𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)(33)

3. Receive next measurement 𝒚(𝑛).4. Compute Jacobian matrix 𝑱∖𝑘(𝑛) for interfering users.for 𝑖 = 1 to 𝑁𝑝 do

(c) Compute the conditional channel estimate 𝒇𝑖(𝑛∣𝑛−1) with covari-ance 𝑷𝑘,𝑘,𝑖(𝑛∣𝑛− 1) conditioned on 𝜖𝑛−1

𝑖 .for 𝑚 = 1 to 2𝑀 + 1 do

(d) Define a grid: 𝛿𝑚 = Δ𝜖(−𝑀 +𝑚− 1).(e) Compute 𝜋𝑖(𝑚) ∝ 𝒩 (𝒚(𝑛);𝒚(𝑖,𝑚)(𝑛∣𝑛 − 1),Σ(𝑖,𝑚)(𝑛∣𝑛 −1))𝒩 (𝛿𝑚;𝛼𝑞

𝑘,𝜖𝜖𝑖(𝑛− 1), 𝜂𝑞𝑘,𝜖) according to (28).end for(f) Find �̃� with a random variable 𝑢 = 𝑟𝑎𝑛𝑑 satisfying

�̃�∑

𝑙=−𝑀


> 𝑢 >

�̃�−1∑

𝑙=−𝑀


. (34)

(g) Update a trajectory 𝜖𝑛𝑖 = {𝛿�̃�, 𝜖𝑛−1𝑖 }.

(h) Update 𝑷∖𝑘,𝑘(𝑛∣𝑛) using eq. (18).(i) Compute channel estimate and its error covariance matrix

𝑷𝑘,𝑘,𝑖(𝑛∣𝑛) = ℬ𝑖𝑷𝑘,𝑘,𝑖(𝑛∣𝑛− 1)(ℬ𝑖)𝑇 −ℬ𝑖𝑷𝑘,∖𝑘(𝑛∣𝑛)(𝒞𝑖)𝑇

−𝒞𝑖𝑷∖𝑘,𝑘(𝑛∣𝑛)(ℬ𝑖)𝑇 + 𝒞𝑖𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1)(𝒞𝑖)𝑇

+𝑁0

𝑇𝑠𝑲𝑖(𝑛)𝑲𝑖(𝑛)

𝑇 ,

𝒇𝑖(𝑛∣𝑛) = 𝒇𝑖(𝑛∣𝑛− 1) +𝑲𝑖(𝑛)(𝒚(𝑛) − 𝒚𝑖(𝑛∣𝑛− 1)),

ℬ𝑖 = 𝑰 −𝑲𝑖(𝑛)ℍ(𝜖𝑖(𝑛)),

𝒞𝑖 = 𝑲𝑖(𝑛)𝑱∖𝑘(𝑛), (35)

where the Kalman gain is defined in (31).(j) Compute importance weight 𝛽𝑖(𝑛) according to (25) and (26).

end for5. Compute the unconditional channel and frequency offset estimates.

𝒇(𝑛∣𝑛) =𝑁𝑝∑

𝑖=1

𝛽𝑖(𝑛)𝒇𝑖(𝑛∣𝑛) and 𝜀(𝑛) ≈𝑁𝑝∑

𝑖=1

𝛽𝑖(𝑛)𝜖𝑖(𝑛). (36)

6. Compute the unconditional error covariance for user 𝑘

𝑷𝑘,𝑘(𝑛∣𝑛) =𝑁𝑝∑

𝑖=1

𝛽𝑖(𝑛)[(𝒇𝑖(𝑛∣𝑛)𝒇𝑖(𝑛∣𝑛)𝑇

+𝑷𝑘,𝑘,𝑖(𝑛∣𝑛)]− 𝒇(𝑛∣𝑛)𝒇(𝑛∣𝑛)𝑇 . (37)

V. JOINT DATA DETECTION AND PARAMETER

ESTIMATION

To this point, we have assumed that an entire trainingblock is employed for joint channel and frequency offsetestimation. In practical OFDMA systems, such training blocksare only available in the beginning of each frame, and only𝑁𝑠𝑝 < 𝑁 pilot symbols are embedded into each data blockto track the time-variations of CFOs and channels. However,for high-mobility applications, the pilots in each data blockmay not provide satisfactory tracking performance. Recently,an EM-based iterative joint data detection and channel/CFOestimation approach has been proposed in [14], where tentativedata decisions are exploited to improve the channel/CFOestimates in an iterative manner. We next similarly combine

joint data detection based on QRD-M [26], [27] with the SK-APF channel/CFO estimator.

Without loss of generality, we focus on data detectionfor the 𝑘-th user in the sequel. Furthermore, ℐ𝑘,𝑑 and ℐ𝑘,𝑝denote the data and pilot carrier index sets for the 𝑘-th user,respectively. The DFT of the received signal at antenna 𝑞 is

𝒓𝑞(𝑛) = 𝑾Δ(𝜀𝑞𝑘(𝑛))

𝑀𝑡∑𝑝=1

𝑫𝑝𝑘(𝑛)𝒉𝑝,𝑞𝑘 (𝑛)+

𝑾

𝐾∑𝑘′=1,𝑘′ ∕=𝑘

Δ(𝜀𝑞𝑘′(𝑛))

𝑀𝑡∑𝑝=1

𝑫𝑝𝑘′(𝑛)𝒉𝑝,𝑞𝑘′ (𝑛) + 𝒗𝑞(𝑛).

(38)

Consider detection of the 𝑚-th data carrier assigned to the𝑘-th user. In the presence of CFO, the corresponding receivedsignal distorted by ICI can be expressed as

𝑟𝑞𝑘,𝑚∣ℐ𝑘,𝑑(𝑛) = 𝑟𝑞𝑘,𝑠,𝑚∣ℐ𝑘,𝑑

(𝑛) + 𝑟𝑞𝑘,𝑖𝑐𝑖,𝑚(𝑛)

+ 𝑣𝑞𝑘,𝑚∣ℐ𝑘,𝑑(𝑛), (39)

where 𝑟𝑞𝑘,𝑠,𝑚∣ℐ𝑘,𝑑(𝑛) and 𝑟𝑞𝑘,𝑖𝑐𝑖,𝑚(𝑛) are the desired signal and

the ICI signal from other subcarriers, respectively. These termscan be explicitly written as

𝑟𝑞𝑘,𝑠,𝑚∣ℐ𝑘,𝑑(𝑛) =

𝑒𝑗𝜃𝑞𝑘(𝑛)𝜇(𝜀𝑞𝑘(𝑛))

𝑀𝑡∑𝑝=1

𝐻𝑝,𝑞𝑘,𝑚∣ℐ𝑘,𝑑(𝑛)𝑑𝑝𝑘,𝑚∣ℐ𝑘,𝑑

(𝑛),

𝑟𝑞𝑘,𝑖𝑐𝑖,𝑚(𝑛) =

𝐾∑𝑘′=1

𝑒−𝑗𝜃𝑞

𝑘′(𝑛)𝑁−1∑𝑚′=0𝑚′ ∕=𝑚

𝜇(𝜀𝑞𝑘′ (𝑛) +𝑚′ −𝑚)×

𝑀𝑡∑𝑝=1

𝐻𝑝,𝑞𝑘′,𝑚′(𝑛)𝑑𝑝𝑘′,𝑚′(𝑛),

(40)

where 𝜇(𝜃)△= 1𝑁 𝑒𝑗𝜋𝜃(𝑁−1)/𝑁 𝑠𝑖𝑛(𝜋𝜃)

𝑠𝑖𝑛(𝜋𝜃/𝑁) and

𝐻𝑝,𝑞𝑘,𝑚(𝑛)△=(𝑾 )(𝑚,1:𝐿𝑓 )𝒉

𝑝,𝑞𝑘 (𝑛)

△=(𝑾𝐿𝑓

)𝑚𝒉𝑝,𝑞𝑘 (𝑛). Note that

in (40), 𝑟𝑞𝑘,𝑖𝑐𝑖,𝑚(𝑛) includes CFO-induced self-interferenceand MAI [16].

Since the channel and CFO are unknown to the receiver,they are approximated by their predictions. Furthermore, somesubcarriers are employed to modulate pilot symbols for allusers. The estimate of the received signal using the chan-nel/offset predictions to be employed for detection is

𝑟𝑞𝑘,𝑚∣ℐ𝑘,𝑑(𝑛∣𝑛− 1) =

𝑒−𝑗𝜃𝑞𝑘(𝑛)

𝜇(𝜖𝑞𝑘(𝑛∣𝑛− 1))𝑟𝑞𝑘,𝑠,𝑚∣𝐼𝑘,𝑑

(𝑛) ≈

𝑟𝑞𝑘,𝑠,𝑚∣ℐ𝑘,𝑑(𝑛∣𝑛− 1) + 𝑛𝑞𝑘,𝑚∣ℐ𝑘

(𝑛∣𝑛− 1),

(41)

where the desired signal, interferers and noise are modeled as


[28]

𝑟𝑞𝑘,𝑠,𝑚∣ℐ𝑘,𝑑(𝑛∣𝑛− 1) =

𝑀𝑡∑𝑝=1

�̂�𝑝,𝑞𝑘,𝑚∣ℐ𝑘,𝑑(𝑛∣𝑛− 1)𝑑𝑝𝑘,𝑚∣ℐ𝑘,𝑑

(𝑛),

𝑛𝑞𝑘,𝑚∣ℐ𝑘(𝑛∣𝑛− 1) =

𝑒−𝑗𝜃𝑞𝑘(𝑛∣𝑛−1)

𝜇(𝜀𝑞𝑘(𝑛∣𝑛− 1))

(𝑟𝑞𝑘,𝑖𝑐𝑖,𝑚(𝑛∣𝑛− 1) + 𝑣𝑞𝑘,𝑚∣ℐ𝑘,𝑑

(𝑛))

,

𝑟𝑞𝑘,𝑖𝑐𝑖,𝑚(𝑛∣𝑛− 1) =

𝐾∑𝑘′=1

𝑒−𝑗𝜃𝑞

𝑘′ (𝑛∣𝑛−1)𝑁−1∑𝑚′=0𝑚′ ∕=𝑚

𝜇(𝜀𝑞𝑘′ (𝑛∣𝑛− 1) +𝑚′ −𝑚)×

𝑀𝑡∑𝑝=1

�̂�𝑝,𝑞𝑘′,𝑚′(𝑛∣𝑛− 1)𝑑𝑝𝑘′,𝑚′(𝑛).

(42)

In (42), 𝜃𝑞𝑘(𝑛∣𝑛−1) = 2𝜋∑𝑛−1𝑙=0 𝜀

𝑞𝑘(𝑙∣𝑙−1) with 𝜖𝑞𝑘(0∣−1) =

𝜖𝑞𝑘(0). Using time updated samples 𝜖𝑖(𝑛∣𝑛 − 1), we approxi-mate the prediction by 𝜀𝑞𝑘(𝑛∣𝑛−1) ≈ ∑𝑁𝑝

𝑖=1 𝛽𝑖(𝑛)𝜖𝑖(𝑛∣𝑛−1),with indices 𝑞, 𝑘 suppressed in 𝜖𝑖(𝑛∣𝑛− 1).

Proposition 1: The covariance of 𝑛𝑞𝑘,𝑚∣ℐ𝑘(𝑛∣𝑛−1) is given

by

𝜎2𝑛𝑞,𝑘

(𝑛∣𝑛− 1)△=𝐸{∣𝑛𝑞

𝑘,𝑚∣ℐ𝑘(𝑛∣𝑛− 1)∣2} =

2𝑁0

𝑇𝑠∣𝜇(𝜀𝑞𝑘(𝑛∣𝑛− 1))∣2+

1

𝑀𝑡∣𝜇(𝜀𝑞𝑘(𝑛∣𝑛− 1))∣2𝐾∑

𝑘′=1

𝑁−1∑𝑚′=0,𝑚′ ∕=𝑚

∣𝜇(𝜀𝑞𝑘′(𝑛∣𝑛− 1) +𝑚′ −𝑚)∣2×

Tr{(𝑰𝑀𝑡 ⊗ (𝑿𝑊 )𝑚′) �̂�𝑞𝑘′(𝑛∣𝑛− 1)�̂�𝑞

𝑘′(𝑛∣𝑛− 1)𝐻},(43)

where (𝑿𝑊 )𝑚△=(𝑾𝐿𝑓

)𝐻𝑚(𝑾𝐿𝑓)𝑚.

The derivation of Proposition 1 is given in the Appendix.Let 𝑚 ∈ ℐ𝑘,𝑑. According to (41), we have

𝒓𝑘∣𝑚(𝑛∣𝑛− 1)△=

[𝑟1𝑘∣𝑚(𝑛∣𝑛− 1), 𝑟2𝑘∣𝑚(𝑛∣𝑛− 1), . . . , 𝑟𝑀𝑟𝑘∣𝑚(𝑛∣𝑛− 1)]𝑇 ,≈⎡

⎢⎢⎣�̂�1,1

𝑘∣𝑚(𝑛∣𝑛− 1) . . . �̂�𝑀𝑡,1𝑘∣𝑚 (𝑛∣𝑛− 1)

... . . ....

�̂�1,𝑀𝑟

𝑘∣𝑚 (𝑛∣𝑛− 1) . . . �̂�𝑀𝑡,𝑀𝑟

𝑘∣𝑚 (𝑛∣𝑛− 1)

⎤⎥⎥⎦⎡⎢⎢⎣

𝑑1𝑘∣𝑚(𝑛)...

𝑑𝑀𝑡𝑘∣𝑚(𝑛)

⎤⎥⎥⎦+

⎡⎢⎣

𝑛1𝑘,𝑚∣ℐ𝑘

(𝑛∣𝑛− 1)...

𝑛𝑀𝑟𝑘,𝑚∣ℐ𝑘

(𝑛∣𝑛− 1)

⎤⎥⎦ ,

△=�̂�𝑘∣𝑚(𝑛∣𝑛− 1)𝒅𝑘∣𝑚(𝑛) +𝒏𝑘,𝑚∣ℐ𝑘

(𝑛∣𝑛− 1).(44)

In (44), 𝒏𝑘,𝑚∣ℐ𝑘(𝑛) ∼ 𝒩 (𝒏𝑘∣𝑚(𝑛);0, Σ̂𝑘∣𝑚(𝑛∣𝑛−1)) with

Σ̂𝑘,𝑚∣ℐ𝑘(𝑛∣𝑛−1)

△=diag{𝜎2𝑛1,𝑘(𝑛∣𝑛−1), . . . , 𝜎2𝑛𝑀𝑟,𝑘

(𝑛∣𝑛−1)}.This form of the covariance approximates the ICI at differentreceive antennas as independent, and is similar to that of[26] in the detection problem. The main difference is that thenoise terms in (44) no longer have equal variances. To takeadvantage of QRD-M [26], we employ a a diagonal scaling ofthe received/corrected vector 𝒓𝑘∣𝑚(𝑛∣𝑛−1) for better detectionperformance [29]. Let �̂�𝑘∣𝑚(𝑛∣𝑛− 1) be the diagonal square

root of Σ̂𝑘∣𝑚(𝑛∣𝑛 − 1). Premultiplying 𝒓𝑘∣𝑚(𝑛∣𝑛 − 1) by�̂�−1𝑘∣𝑚(𝑛∣𝑛− 1) yields

𝒓𝑘∣𝑚(𝑛∣𝑛− 1)† ≈�̂�−1𝑘∣𝑚(𝑛∣𝑛− 1)�̂�𝑘∣𝑚(𝑛∣𝑛− 1)𝒅𝑘∣𝑚(𝑛) + 𝒆𝑘∣𝑚(𝑛∣𝑛− 1),

(45)

where 𝐸{𝒆𝑘∣𝑚(𝑛∣𝑛− 1)𝒆𝑘∣𝑚(𝑛∣𝑛− 1)𝐻} ≈ 𝑰.Data detection of {𝒅𝑘∣𝑚(𝑛)} can be performed based on

(45) by employing QRD-M. The QRD-M detector approxi-mates the maximum-likelihood decision.

𝒅𝑘∣𝑚(𝑛) = arg min𝒅𝑘∣𝑚(𝑛)∈∣𝑆∣𝑀𝑡∥∥∥𝒓𝑘∣𝑚(𝑛∣𝑛− 1)† − �̂�𝑘∣𝑚(𝑛∣𝑛− 1)−1�̂�𝑘∣𝑚(𝑛∣𝑛− 1)𝒅𝑘∣𝑚(𝑛)

∥∥∥2

,

(46)where 𝒮 denotes the signal constellation. For QRD-M, thecondition 𝑀𝑟 ≥𝑀𝑡 is required, which can be easily satisfiedin practical uplink MIMO-OFDMA systems.

Now let

�̂�𝑘∣𝑚(𝑛∣𝑛−1)−1�̂�𝑘∣𝑚(𝑛∣𝑛−1) = �̂�QR(𝑛∣𝑛−1)�̂�QR(𝑛∣𝑛−1)

be the QR decomposition, where �̂�QR(𝑛∣𝑛 − 1) is a unitarymatrix and �̂�QR(𝑛∣𝑛 − 1) is an upper triangular matrix.Substituting �̂�𝑘∣𝑚(𝑛∣𝑛 − 1)−1�̂�𝑘∣𝑚(𝑛∣𝑛 − 1) = �̂�QR(𝑛∣𝑛 −1)�̂�QR(𝑛∣𝑛− 1) into (46), we have

𝒅𝑘∣𝑚(𝑛)QRD-M ≈arg min

𝒅𝑘∣𝑚(𝑛)∈∣𝑆∣𝑀𝑡

∥∥∥𝒓𝑘∣𝑚(𝑛∣𝑛− 1)†QR − �̂�QR(𝑛∣𝑛− 1)𝒅𝑘∣𝑚(𝑛)∥∥∥2

,

(47)where 𝒓𝑘∣𝑚(𝑛∣𝑛 − 1)†QR = �̂�QR(𝑛∣𝑛 − 1)𝐻𝒓𝑘∣𝑚(𝑛∣𝑛 − 1)†.Using the upper-triangular property of �̂�QR(𝑛∣𝑛 − 1) andcombining the M-algorithm, we can detect MIMO user dataefficiently. The details of the QRD-M algorithm can be foundin [26] and the references therein.

VI. SIMULATION RESULTS

In this section, computer simulations are performed toconfirm the performance of the proposed schemes. The sim-ulated system has 𝑁 = 128 subcarriers. Unless otherwisespecified, information bits are mapped onto uncoded QPSKsymbols through a Gray map and all users experience the samenormalized Doppler shift of 𝐹𝑑𝑇𝑑 = 0.001. Furthermore, weset 𝛼𝑞𝑘,𝜀 = 0.9999 and 𝜂𝑞𝑘,𝜀 = 10−5 ∀𝑘, 𝑞 in (4). The SEKF

is initialized with 𝒇𝑞𝑘 (1∣0) = 0 and 𝑷 𝑞𝑘 (1∣0) = 𝑰, ∀𝑘, 𝑞whereas the SK-APF is initialized with 𝒇𝑞𝑘,𝑖(1∣0) = 0 and𝑷 𝑞𝑘,𝑖(1∣0) = 𝑰, ∀𝑘, 𝑖, 𝑞. In the following, four examples willbe presented. In the first two examples, we focus on the pilot-aided schemes to exploit training blocks only whereas the lastexamples investigate the semi-blind schemes.

Example 1: Pilot-Aided Estimation With Equal User Power

In this example, we consider a system with 𝑀𝑡 =𝑀𝑟 = 4and 𝐾 = 5 users. Each user has equal user power andoccupies 𝑁𝑘 = 12 subcarriers. Figs. 1 and 2 show the channelestimation mean-squared error (MSE) and the average absolute(ABS) CFO estimation error over all users at 𝐸𝑠/𝑁0 of20 dB, respectively. For the proposed pilot-aided SK-APF,


2 4 6 8 10 12 14 16 18 20

10−1

100

Es/N

0 [dB]

MS

E o

f Cha

nnel

Est

imat

ion

SK−APF, Np=12, N

D=10

SK−APF, Np=4, N

D=10

SEKF

Fig. 1. Channel estimation performance of the proposed pilot-aided schemesas a function of 𝐸𝑠/𝑁0 in a system with equal-power users.

2 4 6 8 10 12 14 16 18 2010

−3

10−2

Es/N

0 [dB]

Abs

olut

e F

requ

ency

Offs

et E

stim

atio

n E

rror

SK−APF, Np=12, N

D=10

SK−APF, Np=4, N

D=10

SEKF

Fig. 2. CFO estimation performance of the proposed pilot-aided schemes asa function of 𝐸𝑠/𝑁0 in a system with equal-power users.

𝑁𝑝 = 4, 12 particles are employed. Furthermore, we use2𝑀 + 1 = 21 points to approximate the CFO in the firstsymbol interval. In the remaining symbol intervals, we use2𝑀 +1 = 11 points. Finally, the multipath intensity profile is𝐸{∣(𝒉𝑝,𝑞𝑘 ∣)𝑙∣2} = {0.6, 0.4} ∀𝑝, 𝑞, 𝑘 with 𝑙 = 1, 2 = 𝐿𝑓 .

Inspection of Figs. 1 and 2 indicates that the SK-APFsubstantially outperforms the pilot-aided SEKF in terms ofCFO estimation errors but has similar channel estimationperformance as the SEKF. Furthermore, it is evident fromFig. 2 that the SK-APF provides more accurate CFO estimatesas the number of particles employed, 𝑁𝑝, increases from 4 to12.

Example 2: Pilot-Aided Estimation With Unequal User Power

Next, we consider the same system in Example 1, exceptthat users have unequal signal powers due to the near-far effectwith the first user 10 log(𝐾) dB stronger than the others.Figs. 3 and 4 show the channel estimation MSE and the

2 4 6 8 10 12 14 16 18 20

10−1

100

Es/N

0 [dB]

MS

E o

f Cha

nnel

Est

imat

ion

SK−APF, Np=12, N

D=10

SK−APF, Np=8, N

D=10

SK−APF, Np=4, N

D=10

SEKF

Fig. 3. Channel estimation performance of the proposed pilot-aided schemesfor weaker users as a function of 𝐸𝑠/𝑁0 in a system with unequal-powerusers.

2 4 6 8 10 12 14 16 18 2010

−3

10−2

Es/N

0 [dB]

Abs

olut

e F

requ

ency

Offs

et E

stim

atio

n E

rror

SK−APF, Np=12, N

D=10

SK−APF, Np=4, N

D=10

SEKF

Fig. 4. CFO estimation performance of the proposed pilot-aided schemesfor weaker users as a function of 𝐸𝑠/𝑁0 in a system with unequal-powerusers.

average ABS CFO estimation error over all weaker users(excluding the first user) at 𝐸𝑠/𝑁0 of 20 dB, respectively.Comparison of Figs. 3-4 and Figs.1-2 suggests that the near-fareffect has only marginal impact on the estimation performanceof the proposed schemes.

Example 3: Semi-Blind Estimation With Equal User Power

The semi-blind SK-APF is now employed with one train-ing block followed by data blocks consisting of both pi-lots and data symbols. To approximate the CFO uncertaintyregion, we generate 2𝑀 + 1 = 101 and 11 grid pointsfor the first symbol interval and remaining symbol inter-vals, respectively. Furthermore, 𝑀𝑡 = 𝑀𝑟 = 2, 𝐾 = 2users and 𝑁𝑘 = 32 subcarriers are employed in the sim-ulation. The multipath intensity profile is 𝐸{∣(𝒉𝑝,𝑞𝑘 )𝑙∣2} ={0.5610, 0.2520, 0.1132, 0.0509, 0.0229} for ∀𝑝, 𝑞, 𝑘 with𝑙 = 1, . . . , 5 = 𝐿𝑓 .


0 2 4 6 8 10 12 14 16 18 20

10−3

10−2

10−1

Es/N

0 [dB]

Bit

Err

or R

ate

Nsp

=4,Np=10

Nsp

=8,Np=10

Nsp

=4,Np=40

Nsp

=8,Np=40

Ideal

Fig. 5. Uncoded BER performance of the proposed semi-blind SK-APF asa function of 𝐸𝑠/𝑁0 with 𝑁𝑠𝑝 scattered pilots and 𝑁𝑝 particles.

Fig. 5 shows the uncoded bit error rate (BER) performanceof the proposed methods with different numbers of scatteredpilots, 𝑁𝑠𝑝 = 4, 8 and particles, 𝑁𝑝 = 10, 40. The curvelabeled “Ideal" is obtained with perfect knowledge of channeland frequency offset. It is evident from Fig. 5 that moreparticles result in better BER performance for semi-blindSK-APF whereas more pilots lead to marginal performanceimprovement. Furthermore, inspection of Fig. 5 reveals thatthe semi-blind SK-APF with 𝑁𝑠𝑝 = 8 and 𝑁𝑝 = 40 has about1.5 dB loss w.r.t. the ideal curve at BER of 10−3.

Figs. 6 and 7 show the channel and CFO estimationperformance as a function of OFDM symbol index 𝑛 at𝐸𝑠/𝑁0 of 20dB, respectively. Figs. 6 and 7 suggest thatincreasing the number of particles from 𝑁𝑝 = 10 to 𝑁𝑝 = 40can provide significantly more accurate CFO and channelestimation at the cost of higher computational complexity.Furthermore, inspection of Fig. 6 suggests that the maximumCFO estimation error occurs at 𝑛 = 2. This is due to the factthat the first symbol is solely composed of pilots, and hencethe initial CFO estimate is more accurate. On symbol 𝑛 = 2,the estimator begins to diverge due to data errors, but witha sufficient number of measurements the CFO error beginsdecreasing on symbol 𝑛 = 3.

Example 4 : Semi-Blind Estimation for Systems With MoreUsers

In this last example, we investigate a system with 𝑀𝑡 =𝑀𝑟 = 4 using the SK-APF. The same parameters are em-ployed to generate frequency offset samples as shown in theprevious example. Figs. 8 and 9 show the BER performancefor 𝐾 = 2 and 𝐾 = 4, respectively. Clearly, more MAI isinduced by the presence of more users in the system. As aresult, the system performance degrades as 𝐾 increases from2 to 4, which is evidenced by comparing the curves in Figs. 8and 9 obtained with 𝑁𝑠𝑝 = 4 scattered pilots and 𝑁𝑝 = 40particles. More specifically, Fig. 9 indicates that semi-blindSK-APF with 𝑁𝑠𝑝 = 4 and 𝑁𝑝 = 40 entails 3.5 dB loss w.r.t.the ideal curve at BER of 10−4. Furthermore, Fig. 9 shows

2 4 6 8 10 12 14

10−3

10−2

OFDM symbol index (n)

Abs

olut

e F

requ

ency

Offs

et E

stim

atio

n E

rror

N

sp=4,N

p=10

Nsp

=8,Np=10

Nsp

=4,Np=40

Nsp

=8,Np=40

Fig. 6. CFO estimation performance of the proposed semi-blind SK-APF asa function of symbol index 𝑛 at 𝐸𝑠/𝑁0 = 20 dB.

2 4 6 8 10 12 14

0.01

0.02

0.03

0.04

0.05

0.06

0.07

OFDM symbol index (n)

MS

E o

f Cha

nnel

Est

imat

ion

N

sp=4,N

p=10

Nsp

=8,Np=10

Nsp

=4,Np=40

Nsp

=8,Np=40

Fig. 7. Channel estimation performance of the proposed semi-blind SK-APFas a function of symbol index 𝑛. at 𝐸𝑠/𝑁0 = 20 dB.

that only marginal performance improvement can be obtainedby increasing the number of particles beyond 𝑁𝑝 = 40.

VII. CONCLUSION

Pilot-aided and semi-blind joint data detection and fre-quency offset/channel estimation schemes have been pro-posed for the uplink MIMO-OFDMA systems. The proposedschemes employ the parallel Schmidt Kalman filter to de-compose the multiuser estimation problem into more tractablesubproblems, each of which deals with only one desired user.Following the decomposition, the Schmidt Rao-Blackwellizedparticle filter is employed to track the time varying channeland CFO of the desired user in each subproblem. Simulationresults have shown that the resulting scheme can provide ac-curate CFO and channel estimates at affordable computationalcomplexity.

Throughout this work, pilot design has not been takeninto account. However, as shown in [6], [30], [31], optimallydesigned pilots can substantially improve the system perfor-


0 2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

Es/N

0 [dB]

Bit

Err

or R

ate

Np=10, N

sp=4

Np=20, N

sp=4

Np=40, N

sp=4

Np=10, N

sp=8

Np=40, N

sp=8

Exact Channel Information

Fig. 8. Uncoded BER performance of the SK-APF as a function of 𝐸𝑠/𝑁0

with 𝑀𝑡 = 𝑀𝑟 = 4, 𝑁𝑠𝑝 = {4, 8} and 𝐾 = 2.

0 2 4 6 8 10 12 14 16 18 20

10−5

10−4

10−3

10−2

10−1

Es/N

0 [dB]

Bit

Err

or R

ate

Np=10

Np=20

Np=40

Np=60

Exact Channel Information

Fig. 9. Uncoded BER performance of the SK-APF as a function of 𝐸𝑠/𝑁0

with 𝑀𝑡 = 𝑀𝑟 = 4, 𝑁𝑠𝑝 = 4 and 𝐾 = 4.

mance. It is anticipated that the performance of both SEKF andSK-APF estimators will improve with such optimized pilotsand this should be investigated in actual system applications.

APPENDIX: PROOF OF PROPOSITION 1

This Appendix summarizes the proof of Proposition 1. For

simplicity, define 𝜆1△=𝜀𝑞𝑘(𝑛∣𝑛 − 1) +𝑚′ −𝑚. Recalling that

users’ symbols are assumed i.i.d., we have

𝜎2𝑛𝑞,𝑘,𝑚(𝑛)

△=𝐸{∣𝑛𝑞𝑘,𝑚∣ℐ𝑘

(𝑛)∣2} =1

∣𝜇(𝜀𝑞𝑘(𝑛∣𝑛− 1))∣2×

𝐸

⎧⎨⎩𝐾∑𝑘′=1

𝑁−1∑𝑚′=0,

𝑚′ ∕=𝑚

∣𝜇(𝜆1)∣2[𝑑1𝑘′,𝑚′(𝑛), . . . , 𝑑𝑀𝑡

𝑘′,𝑚′(𝑛)]𝒂𝑘′×

𝒂𝐻𝑘′ [𝑑1𝑘′,𝑚′(𝑛), . . . , 𝑑𝑀𝑡

𝑘′,𝑚′(𝑛)]𝐻}+ 2𝑁0/𝑇𝑠,

(A.1)

where

𝒂𝑘′△=

⎡⎢⎣

(𝑾𝐿𝑓)𝑚′ �̂�1,𝑞

𝑘′ (𝑛∣𝑛− 1)...

(𝑾𝐿𝑓)𝑚′ �̂�𝑀𝑡,𝑞

𝑘′ (𝑛∣𝑛− 1)

⎤⎥⎦ =

(𝑰𝑀𝑡 ⊗ (𝑾𝐿𝑓)𝑚′)�̂�𝑞𝑘′ (𝑛∣𝑛− 1).

(A.2)

Using (A.2), we have (A.3), see next page, where we haveexploited the following equalities

𝐸{∣∣𝒅𝑝𝑘,𝑚∣ℐ𝑘(𝑛)∣∣2} = 1/𝑀𝑡, (A.4)

(𝑨⊗𝑩)𝐻(𝑪 ⊗𝑫) = (𝑨𝐻𝑪)⊗ (𝑩𝐻𝑫). (A.5)

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TABLE ISUMMARY OF SYMBOLS FOR USER 𝑘 AND RECEIVING ANTENNA 𝑞

Symbol Definition𝑝,𝑞 indices for transmit and receive antennas, respectively

ℐ𝑘, ℐ𝑘,𝑑,ℐ𝑘,𝑝 indices for subcarrier, data symbols and pilot symbols, respectively𝑾 DFT matrix

𝒉𝑝,𝑞𝑘 (𝑛) complex-valued channel vectors between transmit antenna 𝑝 and receive antenna 𝑞

𝜀(𝑛)△=𝜀𝑞𝑘(𝑛) normalized frequency offset

Δ(𝜀𝑞𝑘(𝑛)) frequency offset matrix for one OFDM symbol𝒅𝑝𝑘(𝑛), 𝒅

𝑝𝑘(𝑛) time- and frequency-domain data symbols from antenna 𝑝, respectively

𝑫𝑝𝑘(𝑛), �̃�

𝑝𝜀,𝑘(𝑛) circulant data matrix without and with frequency offset, respectively

𝛼𝑞𝑘,𝜀, 𝛼

𝑞𝑘,ℎ first-order dynamic model coefficients

ℍ(𝜀(𝑛))△=ℍ

𝑞𝑘(𝜀

𝑞𝑘(𝑛)),ℍ∖𝑘(𝜀∖𝑘(𝑛)) real-valued equivalent channel matrix, interfering matrix

𝒇(𝑛), 𝒇∖𝑘(𝑛) real-valued channel vector, interfering channel vector, respectively𝒙(𝑛),𝒙∖𝑘(𝑛) essential state vector, interfering nuisance state vector

𝑨,𝑨∖𝑘 Kalman dynamic matrix, interfering Kalman dynamic matrix𝑸,𝑸∖𝑘 Covariance matrix for dynamic equation, interfering covariance matrix

𝑱(𝑛),𝑱∖𝑘(𝑛) Jacobian matrix, interfering Jacobian matrix𝑷𝑘,𝑘(𝑛∣𝑛− 1),𝑷∖𝑘,𝑘(𝑛∣𝑛− 1),𝑷∖𝑘,∖𝑘(𝑛∣𝑛− 1) Kalman covariance matrices

𝑲(𝑛),𝑲𝑘,SKF(𝑛) Kalman gain matrix, Schmidt-Kalman gain matrix�̂�(𝑛∣𝑛− 1), �̂�(𝑛∣𝑛) Kalman state prediction, Kalman state estimation

𝜖𝑛𝑖 𝑖-th trajectory of the frequency offset particles𝒚𝑛 cumulative observation sequences

𝜋(.), 𝛽𝑖(𝑛) sampling density, 𝑖-th importance sampling weight𝛿𝑚 𝑚-th grid point

𝒇𝑖(𝑛− 1∣𝑛− 1) Kalman channel estimation conditioned on 𝜖𝑛−1𝑖

𝑷𝑘,𝑘,𝑖(𝑛− 1∣𝑛− 1),𝑷𝑘,𝑘,(𝑖,𝑚)(𝑛− 1∣𝑛− 1) Kalman covariance matrices conditioned on 𝜖𝑛−1𝑖

𝑲𝑖(𝑛) Schmidt-Kalman gain𝜋𝑖(⋅) sampling density

𝑀𝑡∣𝜇(𝜀𝑞𝑘(𝑛∣𝑛− 1))∣2𝜎2𝑛𝑞,𝑘,𝑚(𝑛)− 2𝑀𝑡𝑁0

𝑇𝑠=

𝐾∑𝑘′=1

𝑁−1∑𝑚′=0,

𝑚′ ∕=𝑚

∣𝜇(𝜆1)∣2×

Tr((𝑰𝑀𝑡 ⊗ (𝑾𝐿𝑓

)𝑚′)�̂�𝑞𝑘′ (𝑛∣𝑛− 1)�̂�𝑞𝑘′(𝑛∣𝑛− 1)𝐻(𝑰𝑀𝑡 ⊗ (𝑾𝐿𝑓)𝑚′)𝐻

),

=

𝐾∑𝑘′=1

𝑁−1∑𝑚′=0,

𝑚′ ∕=𝑚

∣𝜇(𝜆1)∣2 × Tr([𝑰𝑀𝑡 ⊗ (𝑿𝑊 )𝑚′ ]{�̂�𝑞𝑘′(𝑛∣𝑛− 1)�̂�𝑞𝑘′(𝑛∣𝑛− 1)𝐻}

),

(A.3)

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Kyeong Jin Kim received the M.S. degree fromthe Korea Advanced Institute of Science and Tech-nology (KAIST) in 1991 and the M.S. and Ph.D.degrees in electrical and computer engineering fromthe University of California, Santa Barbara in 2000.During 1991-1995, he was a research engineer at thevideo research center of Daewoo Electronics, Ltd.,in Korea. In 1997, he joined the Data Transmis-sion and Networking Laboratory at the Universityof California, Santa Barbara. After receiving hisdegrees, he joined the Nokia Research Center in

Dallas as a senior research engineer. From 2005 to 2009, he worked withthe Nokia Corporation in Dallas as a L1 specialist. His research has focusedon transceiver design, resource management, and scheduling in the wirelesscommunication systems.

Man-On Pun (M’06) received the BEng. (Hon.) de-gree in electronic engineering from the Chinese Uni-versity of Hong Kong in 1996, the MEng. degree inComputer Science from the University of Tsukuba,Japan in 1999, and the Ph.D. degree in electricalengineering from the University of Southern Cal-ifornia, Los Angeles, in 2006, respectively. He isa research scientist at Mitsubishi Electric ResearchLaboratories (MERL), Cambridge, MA. He heldresearch positions at Princeton University, Princeton,NJ from 2006 to 2008 and Sony Corporation in

Tokyo, Japan, from 1999 to 2001. Dr. Pun’s research interests are in theareas of statistical signal processing and wireless communications. Amonghis publications in these areas is the recent book Multi-Carrier Techniquesfor Broadband Wireless Communications: A Signal Processing Perspective(Imperial College Press, 2008).

Dr. Pun received the best paper award - runner-up from the IEEEConference on Computer Communications (Infocom), Rio de Janeiro, Brazilin 2009; the best paper award from the IEEE International Conference onCommunications, Beijing, China in 2008; and the IEEE Vehicular TechnologyFall Conference, Montreal, Canada in 2006. He is a recipient of severalscholarships including the Japanese Government (Monbusho) Scholarship, theSir Edward Youde Memorial fellowship for Overseas Studies and the Croucherpostdoctoral fellowship.

Ronald A. Iltis received the B.A. (biophysics) fromThe Johns Hopkins University in 1978, the M.Sc inengineering from Brown University in 1980, and thePh.D in electrical engineering from the Universityof California, San Diego in 1984. Since 1984, hehas been with the University of California, SantaBarbara, where he is currently a Professor in theDepartment of Electrical and Computer Engineer-ing. His current research interests are in OFDM,underwater acoustic communications, MIMO radar,and nonlinear estimation. He has also served as a

consultant to government and private industry in the areas of adaptive arrays,neural networks, and spread-spectrum communications.

joint carrier frequency offset and channel estimation for ...(ofdma) is a leading technology for...

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